> 1. If you were a gifted child, what were the activities, games, questions, or ideas that you most loved, or that ignited and really engaged your interest in science and technology?
LEGO. I also learned to read before school age, and enjoyed reading popular science books.
> 1a. Was there any one person who had a disproportionately positive impact on your STEM education, and if so, why and how?
Adult people who were willing to discuss math seriously with me. I was lucky, there was more than one.
To be more precise, the rare combination is finding an adult who is (1) willing to talk to a child, and (2) actually understands math. Just being nice is not enough, if the child already knows that the answers are wrong. Seriously, saying "I don't know" is much better than giving a bullshit answer. If you don't know math, and don't know anyone who does, then maybe a local library contains a useful book. Actually, these days you have Khan Academy, which can guide your child in math across the elementary and high school, highly recommended.
> 2. In what areas will science have the potentially greatest impact for the benefit of humanity in 20 years? What are the biggest and most challenging problems to solve just over the horizon?
Machine learning, biotechnology? But if you have a four years old child, start with building solid foundations of math, programming, physics, chemistry, biology. With good foundations, you can later become good at the latest hype, so you don't need to make a good prediction 20 years in advance.
Sorry, you wanted more specific. But I can't give you a specific answer for 20 years in the future, I am not a prophet. And focusing on the hype before knowing the fundamentals feels wrong to me: it reminds me about all those people who have "their own opinion about theory of relativity and quantum physics" while making elementary mistakes about high-school physics. Also, the more fresh research, the more likely it is to be wrong; the scientific ideas become more reliable as they survive a few years.
> My four year old daughter is exceptionally gifted (generally), and profoundly so with visual-spatial intelligence and math.
You will probably meet a lot of IQ denialists; it is a quite mainstream belief. But it does not matter. The important thing is to allow your daughter to learn at her own speed. (It would be great if all kids got this chance, whether gifted or not. But the school system requires the child to adapt to the speed of curriculum, not the other way round.) Just not slowing her down will already make a difference. (I am surprised to meet many parents who prevent their kids from learning things the kids are interested in, to avoid possible "problems at school" later. This makes me quite sad how the educational system became a brake on knowledge.) Give her the resources, give her the attention when she is curious. Keep it fun. There are two ways to ruin fun: one is pushing children forward faster than is comfortable for them (which changes the fun into a chore), the other is to ignore the progress they made (which sends a signal that doing math is somehow socially inappropriate).
> We want to foster an environment that constantly challenges her, no matter where her interests lead (whether STEM or something else).
For starters, Khan Academy and books will do great. At some later age, it would be useful to find other kids sharing the same interest. (Here you don't need to consider giftedness explicitly; other kids who e.g. do math olympiad will most likely be similar.) Gifted kids who can't find other gifted kids sometimes become friends with older kids instead. (This is another horrible thing about our school system that it filters your social environment strictly by biological age, thus preventing this coping mechanism.)